Numbered Partitions

Definition

Given a set $S$, a numbered partition of $S$ (or labeled partition) is a numbering of some partition of a $S$. In other words, a numbered partition $(\pi _1, \dots , \pi _p)$ of $S$ is such that

\[ \set{\pi _1, \dots , \pi _p} \quad \text{ partitions } \quad S \]

As before, $\pi _i$ is called a part of the partition, for $i = 1, \dots , n$. In this case $p$ is the size and $(\num{\pi _1}, \dots , \num{\pi _p})$ is the shape of the numbered partition.

When we speak simultaneously of partitions (as defined in Partitions) and numbered partitions, we sometimes call a partition an unnumbered partition (or unlabeled partition, allocation). In this case, the size of a partition $P$ of $S$ is the multiset $m: \N \to \N $ defined so that $m(k)$ is the number of parts of size $k$ in $P$.